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Happy 150th birthday to the Riemann Hypothesis - the most famous unsolved problem in mathematics

It has been 150 years since the mathematician Bernhard Riemann published the conjecture which is now one of the most important unsolved problems in mathematics. The Riemann hypothesis encapsulates humankind's attempt to understand the mysteries of the primes: why there is no apparent pattern in the way the primes are
distributed on the number line. The hypothesis is one of the Clay Mathematics Institute's Millennium Prize Problems — anyone who proves (or disproves) it will receive one million dollars.

Maths in a minute - combinatorics

And while we're on the topic of probability, let's answer one of those important mathematical question: how likely are you to win the lottery?

In the UK lottery you have to choose 6 numbers out of 49, and for a chance at the jackpot you need all of your 6 numbers to come up in the main draw. So the question is really how many possible combinations of 6 numbers can be drawn out of 49? There are 49 possibilities for the first number, 48 for the second, and so on to 44 possibilities for the sixth number, so there are 49 x 48 x 47 x
46 x 45 x 44 = 10068347520 ways of choosing those six numbers... in that order. But we don't care which order our numbers are picked, and the number of different ways of picking 6 numbers are 6 x 5 x 4 x 3 x 2 x 1 = 6! = 720. Therefore our six numbers are one of 49 x 48 x 47 x 46 x 45 x 44 / 6! = 13983816 so we have about a one in 14 million chance of hitting the jackpot.
Hmmm...

But on a brighter note, we have just discovered a very useful mathematical fact: the number of combinations of size k (sets of objects in which order doesn't matter) from a larger set of size n is n! / (n-k)! / k!.

This sort of argument lies at the heart of combinatorics, the mathematics of counting. It might not help you win lotto, but it might keep you healthy. It is used to understand how viruses such as influenza reproduce and mutate, by assessing the chances of creating viable viruses from random recombination of genetic segments.

You can read more on combinatorics, including money (lotto), love (well kissing frogs) and fun (juggling and rubiks cubes) on Plus.

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What are the chances of winning the lottery? How much of a football team's league position is due to luck and how much is due to skill? What are the chances of a false positive test result in security or medical screening? Which newspaper headlines are telling the truth? Can you spot a scam before you fall for it?

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Maths Inspiration event in Cambridge — November 24th

There are still places left at the Maths Inspiration morning event in Cambridge on 24th November, organised by Rob Eastaway. All Maths Inspiration events are aimed at sixth formers and more able Year 11 students.

The Cambridge events will be held in West Road Concert Hall on 24th November 2009, 9.45am - 12.30, repeated 1pm - 3.45pm. Speakers include Claire Ellis, Professor Chris Budd, Dr Hugh Hunt and Professor David Spiegelhalter, with talks including What have mathematicians ever done for us?, The Maths of breakfast, and The subtle science of uncertainty.

Tickets are £6 per attendee at all events with one teacher entitled to free entry for every ten students.

For full details of all events and to book tickets, please see the Maths Inspiration website, or contact mail@mathsinspiration.com.

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What would you like to know about your Universe — The last online poll

This poll is now closed. The most popular question was: "How long is a day?" We will publish the answer in an article on Plus shortly. Thank you for taking part!

This is the last online poll in our series to celebrate the International Year of Astronomy 2009. Choose your favourite question from the list on the right, and we'll put the one that proves most popular to world-leading astronomers and cosmologists. The poll will remain open for a month and the answer will be published in a Plus article and
possibly a podcast soon after.

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Cancer is one of the major causes of death in the world (particularly the developed world), with around 11 million people diagnosed and around 7 million people dying each year. The World Health Organisation predicts that current trends show around 9 million will die in 2015, with the number rising to 11.5 million in 2030.

Cancer is the focus of much medical research, but perhaps surprisingly, mathematical research is also playing its part. Mathematician Mark Chaplain and an interdisciplinary team at the University of Dundee, have been awarded 1.7 million euros to develop a virtual model of cancer growth and spread.